Asymptotic Inversion of Dirichlet Series with Applications to the Distribution of Prime Numbers

Authors

  • George Athanasiou Department of Electrical and Electronics Engineering, University of West Attica, Aigaleo, Greece Author
  • Pericles Papadopoulos Department of Electrical and Electronics Engineering, University of West Attica, Aigaleo, Greece Author https://orcid.org/0000-0002-8268-253X
  • Konstantinos Kalkanis Department of Electrical and Electronics Engineering, University of West Attica, Aigaleo, Greece Author https://orcid.org/0000-0002-2336-9068
  • Constantinos Psomopoulos Department of Electrical and Electronics Engineering, University of West Attica, Aigaleo, Greece Author https://orcid.org/0000-0001-6139-1357

Keywords:

Prime numbers, Number theory, Arithmetic functions

Abstract

This paper investigates the inversion of Dirichlet series, and its applications to the distribution of prime numbers. Several techniques will be developed for inversion using integral kernels, conformal mappings, and asymptotic approximations as arguments approach infinity. These methods are applied to key arithmetic functions, including the prime characteristic function, and prime pair counting function. This paper also analyzes prime gaps, prime k-tuples, and divisor functions, providing asymptotic results and error estimates under the Riemann Hypothesis. The presented techniques are generalizable, offering new insights into prime number behavior and related functions.

References

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Published

12-07-2025

Issue

Vol. 1 (2025)

Section

Articles